Integrability conjecture for split tangent bundles on rationally connected manifolds

Let $X$ be a projective manifold with a splitting

$$T_X=V_1\oplus V_2.$$

Assume that $X$ is rationally connected. Integrability conjecture. The subbundles $V_1$ and $V_2$ are integrable. The conjecture concerns whether Beauville-type counterexamples can occur on rationally connected manifolds; the surrounding discussion notes that, once one direct factor is integrable, both factors are integrable with algebraic leaves.